Stefano Bistarelli, Andrea Formisano (Eds.)
ICTCS’14
Fifteenth Italian Conference on Theoretical Computer Science
Perugia, Italy, September 17–19, 2014
Proceedings
Copyright c2014 for the individual papers by the papers’ authors. Copying permitted for private and academic purposes. Re-publication of material from this volume requires permission by the copyright owners.
Editors’ address:
Universit`a di Perugia
Dipartimento di Matematica e Informatica Via Vanvitelli 1
06123 Perugia, Italy
{stefano.bistarelli|andrea.formisano}@unipg.it
Preface
This volume contains the papers presented at ICTCS 2014, the 15th Italian Conference on Theoretical Computer Science held on September 17-19, 2014 in Perugia.
ICTCS is the traditional meeting of the Italian Chapter of the European Association for Theoretical Computer Science (EATCS). The purpose of these meetings is fostering the cross-fertilisation of ideas stemming from different areas of theoretical computer science.
Hence, they represent occasions for exchanging ideas and for sharing experiences between researchers. They also provide the ideal environment where junior researchers and PhD students can meet senior researchers.
The Italian Chapter of the EATCS was founded in 1972 and previous meetings took place in Pisa (1972), Mantova (1974 and 1989), L’Aquila (1992), Ravello (1995), Prato (1998), Torino (2001), Bertinoro (2003), Pontignano (2005), Roma (2007), Cremona (2009), Camerino (2010), Varese (2012) and Palermo (2013). As usual, ICTCS 2014 was open to researchers from outside Italy, who are always welcome to submit papers and attend these periodical events.
In this edition, there were 30 submitted contributions. Each of them was reviewed by at least 3 Program Committee members. The Committee decided to accept 26 papers covering several areas of theoretical computer science. The participants came from in- stitutions of various countries, namely, China, Finland, France, India, Israel, Italy, Japan, Poland, Tunisia, Turkey, UK, and USA. The program included two invited speakers, Rocco De Nicola (IMT, Lucca) and Giuseppe Liotta (Universit`a di Perugia) and a presentation given by Flavio Chierichetti (Sapienza Universit`a di Roma), the recipient of theYoung Researcher in Theoretical Computer Science Award 2014, conferred this year by the Ital- ian Chapter. Furthermore, Livio Bioglio (INSERM, Paris) and Andrea Marino (Universit`a di Milano), the two recipients of theBest PhD Thesis in Theoretical Computer Science Award 2014, assigned by the Italian Chapter, gave two talks illustrating their recent re- search. The program of ICTCS 2014 included a special session devoted to the memory of Alberto Bertoni, which was one of the founders of the Italian Chapter and recently passed away. This session, was organized by Arturo Carpi and Alessandra Cherubini.
We would like to express our gratitude to the invited speakers, to the recipients of the three Awards, and to all authors and participants. We also wish to thank the members of the Program Committee and all additional anonymous reviewers for their hard work. A special mention is due to the colleagues of the Organizing Committee for the invaluable contribution they gave in organizing ICTCS 2014.
We would like to give special thanks to the various sponsors that supported the event:
EATCS, Universit`a di Perugia, Dipartimento di Matematica e Informatica, Regione Um- bria, Provincia di Perugia, Comune di Perugia, Fondazione Perugiassisi 2019, Fondazione Cassa di Risparmio di Perugia, INdAM-GNCS, IOS Press. Finally, we mention Easy- Chair and CEUR-WS.org that helped us in organizing the conference and producing the proceedings.
September 2014
Perugia Stefano Bistarelli
Andrea Formisano
Program Committee
Paolo Baldan Universit`a di Padova Giampaolo Bella Universit`a di Catania Marco Bernardo Universit`a di Urbino Davide Bilo Universit`a di Sassari Stefano Bistarelli (chair) Universit`a di Perugia Michele Boreale Universit`a di Firenze Tiziana Calamoneri Sapienza Universit`a di Roma Antonio Caruso Universit`a del Salento Ferdinando Cicalese Universit`a diSalerno Flavio Corradini Universit`a di Camerino Giorgio Delzanno Universit`a di Genova Mariangiola Dezani Universit`a di Torino Eugenio Di Sciascio Politecnico di Bari Agostino Dovier Universit`a di Udine
Marco Faella Universit`a di Napoli “Federico II”
Michele Flammini Universit`a di L’Aquila Andrea Formisano (chair) Universit`a di Perugia Maurizio Gabbrielli Universit`a di Bologna Fabio Gadducci Universit`a di Pisa Raffaella Gentilini Universit`a of Perugia
Laura Giordano Universit`a del Piemonte Orientale Giuseppe F. Italiano Universit`a di Roma ”Tor Vergata”
Sabrina Mantaci Universit`a di Palermo Isabella Mastroeni Universit`a di Verona
Manuela Montangero Universit`a di Modena e Reggio Emilia Maurizio Proietti IASI-CNR, Roma
Antonino Salibra Universit`a Ca’ Foscari Venezia Francesco Santini Universit`a di Perugia
Marinella Sciortino Universit`a di Palermo Maurice Ter Beek ISTI-CNR, Pisa
Local Organizing Committee
Serena Arteritano Stefano Bistarelli Arturo Carpi Andrea Formisano Raffaella Gentilini Bruno Iannazzo Laura Marozzi Alfredo Navarra
Fernanda Pambianco Fabio Rossi
Francesco Santini Simone Topini Lidia Trotta Emanuela Ughi Flavio Vella
Contents
Invited Talks
A formal approach to autonomic systems programming: the SCEL language
Rocco De Nicola 1
Graph drawing beyond planarity: some results and open problems
Giuseppe Liotta 3
ICTCS Young TCS Research Award
Trace complexity
Flavio Chierichetti 9
ICTCS Doctoral Research Awards
Type disciplines for systems biology
Livio Bioglio 11
Algorithms for biological graphs: analysis and enumeration
Andrea Marino 15
Regular Papers
Timed process calculi: from durationless actions to durational ones
Marco Bernardo, Flavio Corradini, Luca Tesei 21
Size-constrained 2-clustering in the plane with Manhattan distance
Alberto Bertoni, Massimiliano Goldwurm, Jianyi Lin, Linda Pini 33 Graphs of edge-intersecting and non-splitting paths
Arman Boyacı, Tınaz Ekim, Mordechai Shalom, Shmuel Zaks 45 A graph-easy class of mute lambda-terms
Antonio Bucciarelli, Alberto Carraro, Giordano Favro, Antonino Salibra 59
CONTENTS
Relating threshold tolerance graphs to other graph classes
Tiziana Calamoneri, Blerina Sinaimeri 73
ˇCern´y-like problems for finite sets of words
Arturo Carpi, Flavio D’Alessandro 81
Reasoning about connectivity without paths
Alberto Casagrande, Eugenio Omodeo 93
Binary 3-compressible automata
Alessandra Cherubini, Andrzej Kisielewicz 109
Extendibility of Choquet rational preferences on generalized lotteries
Giulianella Coletti, Davide Petturiti, Barbara Vantaggi 121 On multiple learning schemata in conflict driven solvers
Andrea Formisano, Flavio Vella 133
A metamodeling level transformation from UML sequence diagrams to Coq
Chao Li, Liang Dou, Zongyuan Yang 147
An efficient algorithm for generating symmetric ice piles
Roberto Mantaci, Paolo Massazza, Jean-Baptiste Yun`es 159 Adding two equivalence relations to the interval temporal logicAB
Angelo Montanari, Marco Pazzaglia, Pietro Sala 171
Efficient channel assignment for cellular networks modeled as honeycomb grid Soumen Nandi, Nitish Panigrahy, Mohit Agrawal, Sasthi C. Ghosh, Sandip Das 183 Programmable enforcement framework of information flow policies
Minh Ngo, Fabio Massacci 197
On the Stackelberg fuel pricing problem
Cosimo Vinci, Vittorio Bil`o 213
Structural complexity of multi-valued partial functions computed by nondeterministic pushdown automata
Tomoyuki Yamakami 225
CONTENTS
Communications
Proving termination of programs having transition invariants of heightω
Stefano Berardi, Paulo Oliva, Silvia Steila 237
Orthomodular algebraic lattices related to combinatorial posets
Luca Bernardinello, Lucia Pomello, Stefania Rombol`a 241 Abstract argumentation frameworks to promote fairness and rationality in
multi-experts multi-criteria decision making
Stefano Bistarelli, Martine Ceberio, Joel A. Henderson, Francesco Santini 247 Optimal placement of storage nodes in a wireless sensor network
Gianlorenzo D’Angelo, Daniele Diodati, Alfredo Navarra, Cristina M. Pinotti 259 Engineering shortest-path algorithms for dynamic networks
Mattia D’Emidio, Daniele Frigioni 265
Minimal models for rational closure inSHIQ
Laura Giordano, Valentina Gliozzi, Nicola Olivetti, Gian Luca Pozzato 271 An algebraic characterization of unary two-way transducers
Christian Choffrut, Bruno Guillon 279
Logspace computability and regressive machines
Stefano Mazzanti 285
Papers not included here and published elsewhere
Operational state complexity under Parikh equivalence
Giovanna Lavado, Giovanni Pighizzini and Shinnosuke Seki. Appeared in H. Jur- gensen et al. (Eds.): DCFS 2014, LNCS 8614, pp. 294-305. Springer (2014)
Author Index 291
A formal approach to autonomic systems programming:
the SCEL language
Rocco De Nicola
IMT – Institute for Advanced Studies Lucca rocco.denicola@imtlucca.it
Abstract. The autonomic computing paradigm has been proposed to cope with size, complexity and dynamism of contemporary software-intensive systems. The challenge for language designers is to devise appropriate abstractions and linguis- tic primitives to deal with the large dimension of systems, and with their need to adapt to the changes of the working environment and to the evolving require- ments. We propose a set of programming abstractions that permit to represent behaviours, knowledge and aggregations according to specific policies, and to support programming context-awareness, self-awareness and adaptation. Based on these abstractions, we define SCEL (Software Component Ensemble Lan- guage), a kernel language whose solid semantic foundations lay also the basis for formal reasoning on autonomic systems behaviour. To show expressiveness and effectiveness of SCEL’s design, we present a Java implementation of the pro- posed abstractions and show how it can be exploited for programming a robotics scenario that is used as a running example for describing features and potentials of our approach.
Graph drawing beyond planarity:
some results and open problems
Giuseppe Liotta Dipartimento di Ingegneria Universit`a degli Studi di Perugia, Italy
giuseppe.liotta@unipg.it
Abstract. We briefly review some recent findings and outline some emerging research directions about the theory of “nearly planar” graphs, i.e. graphs that have drawings where some crossing configurations are forbidden.
1 Graph drawing beyond planarity
Recent technological advances have generated torrents of relational data that are hard to display and visually analyze due, mainly, to their large size. Application domains where this need is particularly pressing include Systems Biology, Social Network Analysis, Software Engineering, and Networking. What is required is not simply an incremental improvement to scale up known solutions but, rather, a quantum jump in the sophisti- cation of the visualization systems and techniques. New research scenarios for visual analytics, network visualization, and human-computer interaction paradigms must be identified; new combinatorial models must be defined and their corresponding theoret- ical problems must be computationally investigated; finally, the theoretical solutions must be experimentally evaluated and put into practice. Therefore, a substantial re- search effort in the graph drawing and network visualization communities started from the following considerations.
The Planarity Handicap. The classical literature on graph drawing and network visu- alization showcases elegant algorithms and sophisticated data structures under the assumption that the input relational data set can be displayed as a network where no two edges cross (see, e.g., [14,35,36,40]), i.e. as a planar graph. Unfortunately, almost every graph is non-planar in practice and various experimental studies have established that the human ability of understanding a diagram is dramatically af- fected by the type and number of edge crossings (see, e.g., [42,43,48]).
Combinatorial Topology vs. Algorithmics. A topological graph is a drawing of a graph in the plane such that vertices are drawn as points and edges are drawn as sim- ple arcs between the points. Extremal theory questions such as “how many edges can a certain type of non-planar topological graph have?” have been investigated by mathematicians for decades, typically under the name of Tur´an-type problems.
However, the corresponding computational question: “How efficiently can one com- pute a drawingΓ of a non-planar graph such thatΓ is a topological graph of a cer- tain type?” has been surprisingly disregarded by the algorithmic community until very recent years.
We recall that planar graphs can be expressed in terms of forbidden subgraphs:
A graphGis planar if and only if it does not contain a subdivision of K5 or K3,3. Then, a fundamental natural step towards understanding non-planar graphs is to con- sider network visualizations where some types of crossings are forbidden while some other types are allowed. For example, we recall a sequence of HCI experiments by Huang et al. [32,33,34] proving that crossing edges significantly affect human under- standing if they form acute angle, while crossing that form angles from about π3 to π2 guarantee good readability properties. Hence it makes sense to explore complexity is- sues related to drawings of graphs where such “sharp angle crossings” are forbidden.
As another problem, Purchase et al. [42,43,48]) prove that an edge is difficult to read if it is crossed by many other edges; hence, the current research agenda considers com- putational issues with graph drawings where every edge is crossed by at mostkother edges, for a given constantk.
In addition to requiring that some types of edge crossings must be forbidden, non- planar drawings must also satisfy a set of geometric optimization goals (often called aesthetic requirements) such as, for example, minimizing the area of the drawing for a given resolution rule, maximizing the aspect ratio, minimizing the number of different slopes used to draw the edges, or the number bends along the edges.
In the next section we briefly recall some of the most recent results in the area and propose a few open problems. More formally, adrawingof a graphG:(i)injectively maps each vertexuofGto a pointpuin the plane;(ii)maps each edge(u, v)ofGto a Jordan arc connectingpuandpvthat does not pass through any other vertex;(iii)is such that any two edges have at most one point in common. A drawing of a graph is a straight-line drawingif every edge is a straight-line segment, it is apoly-line drawing if the edges are polygonal chains and may contain bends.
2 Some results and open problems
The “beyond planarity” research area could be briefly described as the (potentially un- countable) collection of problems of the type depicted in Figure 1, where the column
”Forbidden” describes a forbidden crossing configuration and the column ”Question”
describes a corresponding computational question of interest in graph drawing. We re- mark that both the forbidden configurations and the computational questions of Figure 1 are mere examples within a much larger research framework. In the remainder, we only give some references about the second and the fourth entry of the table. The interested reader is referred, for example, to recent proceedings of the International Symposium on Graph Drawing [49] for more results on the “beyond planarity” topic. (See also http://www.graphdrawing.org/symposia.html.)
2.1 Drawings with large crossing angles
The crossing angle resolution of a drawing of a graph measures the smallest angle formed by any pair of crossing edges.
ARAC drawingis a drawing of a graph whose edges can cross only orthogonally to one another, i.e. a RAC drawing maximizes the crossing angle resolution. The no- tion of RAC drawings was first introduced by Didimo et al. in [23], who studied both G.Liotta. Graph drawing beyond planarity: some results and open problems
no crossing angle
no fan crossing α
smaller thanα
no edge with k crossings
with bounded vertex degree Straight−line drawability for graphs edges
no three mutually crossing
Compute compact drawings of planar graphs with lmited number of crossings per edge
of graph pairs
Compute simultaneous emebddings Complexity of the recognition problem
Forbidden configuration Algorithmic question
Fig. 1.A table with some forbidden crossing configurations and related computational questions.
straight-line and poly-line drawings. Variants of RAC drawings are drawings in which the minimum crossing angle must be at least a given constat or the drawings where the minimum crossing angle is exactly a given constant. A limited list of recent pa- pers about RAC drawings and their variants includes [4,5,6,7,15,16,17,18,22,25,47].
See also [24] for more references and open problems about drawing graphs with large crossing angles. A sample open problem follows.
Open Problem:Argyriouet al.[6] prove that deciding whether a graph has a straight- line RAC drawing is NP-hard. Hence, maximizing the crossing angle resolution in a straight-line drawing of a graph is also NP-hard. Is there an efficient approximation algorithm for this problem? Is there a polynomial time solution for special families of graphs (e.g. those having bounded vertex degree)?
Related to the problem above, we recall that there is a polynomial time algorithm to recognize whether a bipartite graph has a straight-line RAC drawing such that the vertices of a same partition set all lie on one of two parallel lines [16].
G.Liotta. Graph drawing beyond planarity: some results and open problems
2.2 Drawings with few crossings per edge
For a fixed non negative integerk, ak-planar drawingis a drawing of a graph where every edge can be crossed by at mostkother edges. Ak-planar graphis a graph that has ak-planar drawing. Note that the family of0-planar graphs coincides with the family of planar graphs. The literature about drawings of graphs where every edge can be crossed at mostktimes has mostly focused on the casek= 1.
Concerning Tur´an-type problems, Pach and T´oth prove that1-planar graphs with nvertices have at most 4n−8edges, which is a tight upper bound [41]; in the case of straight-line drawings, Didimo [21] proved that a tight bound is4n−9.1-planarity testing is studied by Korzhik and Mohar who prove that recognizing1-planar graphs is NP-hard [39]; polynomial-time solutions for the recognition problem are known under some additional assumptions and/or for restricted classes of graphs (see, e.g. [8,27,30]).
Straight-line1-planar drawings have been studied in [3,31,46]. The relation between 1-planar drawings and RAC drawings is considered in [13,28]. A limited list of addi- tional papers on1-planar graphs includes [1,2,3,9,10,11,26,29,31,37,38,45].
We conclude with a classical open problem about trade-offs of different aesthetic requirements. Assuming that the vertices are points of an integer grid, thearea of a drawingof a graph is defined as the area of the smallest axis aligned rectangle that includes the drawing.
Open Problem:It is known that every planar graph withnvertices admits a crossing- free straight-line drawing inΘ(n2)area [12,44]. On the other hand, every planar graph can be drawn with straight-line edges inO(n)area if one allowsO(n)crossings per edge [50]. Does every planar graph withnvertices have a straight-line drawing with o(n2)area and ao(n)crossings per edge?.
Starting references to study the above problem include [19,20].
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G.Liotta. Graph drawing beyond planarity: some results and open problems
Trace complexity
Flavio Chierichetti Dipartimento di Informatica Sapienza University of Rome
Abstract. Prediction tasks in machine learning usually require deduc- ing a latent variable, or structure, from observed traces of activity — sometimes, these tasks can be carried out with a significant precision and statistical significance, while sometimes getting any useful predic- tion requires an unrealistically large number of traces.
In this talk, we will study the trace complexity of (that is,the number of traces needed for carrying out) two prediction tasks in social networks:
the network inference problem and the number of signers problem.
The first problem [1] consists of reconstructing the edge set of a network given traces representing the chronology of infection times as epidemics spread through the network. The second problem’s [2] goal is to guess the unknown number of signers of some email-based petitions, when only a small subset of the emails that circulated is available.
These two examples will allow us to make some general remarks about social networks’ prediction tasks.
References
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2. F. Chierichetti, J. M. Kleinberg, and D. Liben-Nowell. Reconstructing patterns of information diffusion from incomplete observations. In J. Shawe-Taylor, R. S. Zemel, P. L. Bartlett, F. C. N. Pereira, and K. Q. Weinberger, editors,NIPS, pages 792–800, 2011.
Type disciplines for systems biology
Livio Bioglio
INSERM, UMR-S 1136, iPLESP, Paris, France
Systems Biology is a discipline that aims to study complex biological sys- tems by means of computational models. Because of the complexity of biological behaviors, the formalisms used in this field are usually designed ad-hoc for the biological topic of interest, or they need to be tuned by a long set of evolu- tion rules. Here we present a different approach: we define biological properties through a type discipline, leaving the formalisms as general as possible. We ex- plore three different kinds of Type Systems: a static one, that limits the model that can be written by modelers; a dynamic one, that limits the evolution of the model at run-time; and an hybrid combination of the previous ones.
1 Static type system
Homogeneous biological entities are classified according to their behavior. In order to reproduce such classification, we propose a Minimal Object-Oriented Core Calculus for term-rewriting formalisms. A rewrite system is composed by a term, representing the structure of the modeled system, and a series of reduction rules, representing the possible evolutions of the system: depending on the for- malism, these rules can be embedded in terms, like in P Systems [7], or defined in a separate part, like in the Calculus of Looping Sequences [1] (CLS for short).
The objective of the OO Core Calculus is to facilitate the organizations of rules, also improving their re-use, and to check the correctness of the model at compile time. In our core calculus it is possible to define classes, that contain methods (encapsulation) and extend another class (subtyping), inheriting all its methods (inheritance). Methods are formed by a sequence of variables, the arguments, and a sequence of reduction rules containing these variables. They are called on symbols of the model, representing biological entities, with a sequence of values as arguments (method invocation). A method invocation is replaced by the re- duction rules of the method, in which the variables are replaced by the values used as arguments. These reduction rules are then used for the evolution of the model. The syntax, definitions and rules of the OO Core Calculus are inspired by the ones proposed by Igarashi, Pierce and Wadler for Featherweight Java [6], a minimal core calculus for modeling the Java Type System. For more details, see [2].
2 Dynamic type system
In Biology and Chemistry there can be found several examples of repellency, such as hydrophobicity (the physical property of a molecule that is repelled from a
mass of water), the behavior of anions and cations, or, at a different level of abstraction, the behavior of the rh antigen for the different blood types. As a counterpart, there may be elements, in nature, which always require the pres- ence of other elements: for example, it is difficult to find a lonely atom of oxygen, they always appear in the pairO2. We bring these aspects at their maximum limit, and, by abstracting away all the phenomena which give rise/arise to/from repellency (and its counterpart), we assume that for each kind of element of our reality we are able to fix a set of elements which are required by the element for its existence, and a set of elements whose presence is forbidden by the element.
We enrich the basic CLS with a type discipline which guarantees the soundness of reduction rules with respect to some relevant properties of biological systems deriving from the required and excluded kinds of elements. The key technical tool we use is to associate to each reduction rule the minimal set of conditions an instantiation must satisfy in order to assure that applying this rule to a ”correct”
system we get a ”correct” system as well. We also propose a type inference algo- rithm, based on the machinery of principal typing [8], and show its soundness and completeness. The required/excluded elements properties modeled here assure, through type inference, that only compatible elements are combined together in the different environments of the biological system took in consideration. Thus the type system intrinsically yields a notion of correct (well-behaving) system according to the expressed requirements. The detailed Type Discipline can be found in [5].
There are cases in which the request/repellency model cannot reflect the behav- ior of a biological system. An example is homeostasis, the property of a system that regulates its internal environment and tends to maintain stable conditions that are optimal for survival: when this equilibrium is disturbed, built-in regula- tory devices respond in order to restore the balance. Different living organisms employ homeostatic mechanisms to maintain some conditions in specific ranges:
the human body, like in all the warm-blooded animals, maintain a near-constant body temperature using mechanisms such as vasodilation and vasoconstriction;
microorganisms maintain the iron presence above a minimum level to maintain life but up to a maximum level to avoid iron toxicity. For this reason, we pro- pose an extension of the previous Type Disciplines, where we assume that for each element of our system we can fix the minimum and the maximum number of other elements it requires. We enrich CLS with a type discipline and typed reductions that guarantee the soundness of reduction rules with respect to the properties of biological systems deriving from the minimum and the maximum requested numbers of elements, and a type inference algorithm for inferring the type of rewriting rules. Our contribution appeared in [3].
3 Hybrid type system
We present the variant of the Calculus of Looping Sequences with global and local rewrite rules (CLSLR, for short). Global rules are the usual rules of CLS, and they can be applied anywhere in a given term wherever their patterns match L.Bioglio. Type disciplines for systems biology
the portion of the system under investigation, while local rules can only be applied in the compartment in which they are defined. Terms written in CLSLR are thus syntactically extended to contain explicit local rules within the term, on different compartments. Local rules can be created, moved between different compartments and deleted. Having a calculus in which we can model the dynamic evolution of the rules describing the system allows to study emerging properties of complex systems in a more natural and direct way. As it happens in nature, where data and programs are encoded in the same kind of molecular structures, we insert rewrite rules within the terms modeling the system under investigation.
On the other hand, some rule may represent general behaviors, common to the whole system: global rules are used for avoiding the repetition of such rules in each compartment. Since in this framework the focus is put on local rules, we define a set of features that can be associated to each one. Features may define general properties of rewrite rules or properties which are strictly related to the model under investigation. We define a membrane type for the compartments of our model and develop a type systems enforcing the property that a compartment must contain only local rules with specific features. Our framework has been presented in [4].
References
1. R. Barbuti, A. Maggiolo-Schettini, P. Milazzo and A. Troina. A Calculus of Looping Sequences for Modelling Microbiological Systems.Fundamenta Informaticæ, 72(1–
3):21–35, 2006.
2. L. Bioglio. A Minimal OO Calculus for Modelling Biological Systems.Computational Models for Cell Processes (CompMod) 2011, EPTCS 67:50–64, 2011.
3. L. Bioglio. Enumerated Type Semantics for the Calculus of Looping Sequences.
RAIRO - Theoretical Informatics and Applications, 45(01):35–58, 2011.
4. L. Bioglio, M. Dezani-Ciancaglini, P. Giannini and A. Troina. A Calculus of Loop- ing Sequences with Local Rules.7th Workshop on Developments in Computational Models (DCM’11), EPTCS 88:43–58, 2011.
5. L. Bioglio, M. Dezani-Ciancaglini, P. Giannini and A. Troina. Type Directed Se- mantics for the Calculus of Looping Sequences.International Journal of Software and Informatics, to appear.
6. A. Igarashi, B. Pierce and P. Wadler. Featherweight Java: a Minimal Core Calcu- lus for Java and GJ. ACM Transactions on Programming Languages and System, 23:396–450, 2001.
7. G. P˘aun.Membrane Computing. An Introduction. Springer, 2002.
8. J. B. Wells. The Essence of Principal Typings. International Colloquium on Au- tomata, Languages and Programming (ICALP’02), LNCS 2380:913–925, 2002.
L.Bioglio. Type disciplines for systems biology
Algorithms for biological graphs:
analysis and enumeration
?Andrea Marino
Dipartimento di Informatica, Universit`a di Milano, Milano, Italy
The aim of enumeration is to list all the feasible solutions of a given prob- lem satisfying some constraints. Enumeration algorithms are particularly useful whenever the goal of a problem is not clear and all its solutions need to be checked. Since one peculiar property of biological networks is the uncertainty, a scenario in which enumeration algorithms can be helpful is biological network analysis. Modelling biological networks indeed introduce bias: arc dependencies are neglected and underlying hyper-graph behaviours are forced in simple graph representations to avoid intractability. Moreover regulatory interactions between all the biological networks are omitted, even if none of the different biological layers is truly isolated. Last but not least, the dynamical behaviours of biolog- ical networks are often not considered: indeed most of the currently available biological network reconstructions are potential networks, where all the possible connections are indicated, even if edges/arcs and vertices are hardly present all together at the same time. More details about these aspects of the biological networks can be found in [8].
Our Contribution. We have shown four examples of enumeration algorithms that can be applied to efficiently deal with some biological problems modelled by using biological networks: enumerating central and peripheral nodes of a network, enumerating stories, enumerating paths or cycles, and enumerating bubbles. Notice that the corresponding computational problems we define are of more general interest and our results hold in the case of arbitrary networks.
1 Enumerating central and peripheral vertices
Structural analysis allows the identification of important and not important ver- tices within a network and also for this reason has become very popular in many disciplines. In the biological domain, the importance of a vertex can be defined in many different ways. With neighbourhood-based centrality measures, such as degree, the importance of the vertices is inferred from their local connectivity and the more connections a vertex has the more central it is. Closeness, ec- centricity, and shortest path based betweenness relies on global properties of a network, such as distance between vertices.
?The author wants to thank the PhD advisor Pierluigi Crescenzi, the PhD School of Dipartimento di Sistemi e Informatica, Universit`a di Firenze (Italy), and all the coauthors of the papers.
We have focused on the enumeration of the radial and diametral vertices, i.e. vertices that are central and peripheral according to the eccentricity notion of centrality, and on the computation of the radius and diameter of biological networks and of real world graphs in general. The diameter and radius of a graph are respectively the maximum and minimum eccentricity among all its nodes, where the eccentricity of a node xis the distance from x to its farthest node.
Thus, intuitively, the diametral source vertices are the vertices that hardly reach the other ones, the diametral target vertices are the vertices hardly reachable from the other ones, and the radial vertices are the vertices that easily reach all the vertices of the network. In order to calculate the vertices that can be easily reached from any other vertex, it is sufficient to consider the transposed graph.
We have presented thedifubAlgorithm, which is able to list all the diame- tral sources and targets and to compute the diameter of (strongly) connected components of a graph G = (V, E) in time O(|E|) in practice, even if, in the worst case, the complexity isΘ(|V||E|). Analogously, we have presented a new algorithm to list all the central vertices and to compute the radius of (strongly) connected components of a graph inalmost O(|E|) time in practice.
The analysis of real world networks in general, such as citation, collaboration, communication, road, social, and web networks, has attracted a lot of attention.
The fundamental analysis measures have been reviewed in [12]. Moreover the size of these networks has been increasing rapidly, so that in order to study such measures, algorithms able to handle huge amount of data are needed. Since the algorithms available until now were not able to compute diameter and radius in the case of huge real world graphs, the contribution of our algorithms is not just limited to biological networks analysis, but extends also to the analysis of complex networks in general. We thus have shown their effectiveness also for several other kinds of complex networks. More details can be found in the work [5], which has been the generalization of [4, 3]. Our algorithm in [3] has been used to compute the diameter of Facebook Network (721.1M vertices, 68.7G edges, and diameter 41) with just 17 bfses in a popular work ([9], divulged by New York Times on November 22, 2011).
2 Enumerating stories
The problem of enumerating stories was motivated initially by the biological question in [10] related to Metabolic networks, in particular to compound graphs, in which vertices are compounds and there is an arc from a compoundx to a compoundy if there is a metabolic reaction that consumes x and producesy.
A subsetBcorresponds to compounds that have been experimentally identified as having a significantly higher or lower production in a given condition (for instance when an organism is exposed to some stress). The aim is then to extract all the interaction dependencies among the compounds inBwhich do not create cycles but at the same time involve as many compounds as possible. These may require intermediate steps that concern compounds not inB, but the initial and A.Marino. Algorithms for biological graphs: analysis and enumeration
final steps must involve only compounds in B. A solution, that is a possible scenario of metabolic dependencies, is called a(metabolic) story.
A metabolic story has to capture the relationship between the vertices of interest in a way that allows us to define a flow of matter from a set of sources to a set of target compounds. The need for this hierarchy between the compounds led us to consider acyclic solutions. The maximality condition has been added in order to capture all alternative paths between the sources and the targets.
The problem is then to “tell” all possible stories given as input a graphGand a subsetBof the vertices ofG.
We have presented a polynomial algorithm to find one story and an exact but exponential approach for the enumeration problem [1]. This definition is a generalization of a well-known problem which is thefeedback arc set problem.
However, any polynomial-delay algorithm to enumerate feedback arc sets (ex:
[14]) can only be used in some particular instances. Moreover we have shown that finding a story with a specified set of sources or targets is NP-hard.
Our contribution appeared in [1] and its biological application in [11].
3 Enumerating cycles or paths
Studying paths or cycles of biological networks can be useful for several pur- poses. In the case of interaction graphs, such as Gene Regulatory networks, the importance of enumeration has been shown in [7]. These networks are directed, their vertices are genes, and their arcs are signed, where the sign or weight of the arcs indicates the causal relationship between the vertices, such as activation or inhibition. In particular cycles and paths can be useful for studying dependencies among vertices, the steady state and multistationarity of dynamic models.
We have considered the problem of enumerating paths and cycles in the case of undirected graphs. This result can be useful for undirected Protein-Protein Interaction networks, where nodes are proteins and edges are interactions, but in the case of interaction networks in general, our approach neglects the effects of the controls, i.e. the sign and direction of the arcs. In this latter case, the cycles can be enumerated in the underlying undirected graph and a posteriori filtered orad hoc algorithms can be applied. The main question arising from our work, is whether it is possible to extend our result to directed graphs in order to efficiently deal also with this kind of networks.
On the other hand, our contribution is not just restricted to biological undi- rected networks, but extends also to arbitrary undirected graphs. Listing all the paths and cycles in a graph is a classical problem whose efficient solu- tions date back to the early 70s. The best known solution in the literature is given by Johnson’s algorithm [6] and takes O((|C(G)|+ 1)(|E|+|V|)) and O((|Pst(G)|+ 1)(|E|+|V|)) time for a graph G = (V, E), where C(G) and Pst(G) denote respectively the set of cycles and (s, t)-paths inG. However there exists graphs for which this algorithm is not optimal.
We have presented the first optimal algorithm to list all the paths and cy- cles in an undirected graphG. Our algorithm requiresO(|E|+P
c∈C(G)|c|) time A.Marino. Algorithms for biological graphs: analysis and enumeration
and is asymptotically optimal: indeed, Ω(|E|) time is necessarily required to read G as input, and Ω(P
c∈C(G)|c|) time is necessarily required to list the output. Moreover, our algorithm lists all the (s, t)-paths in G optimally in O(|E|+P
π∈Pst(G)|π|) time, observing thatΩ(P
π∈Pst(G)|π|) time is necessarily required to list the output.
Our algorithm exploits the decomposition of the graph into biconnected com- ponents and without loss of generality restricts to study paths and cycles in a same biconnected component. Thus it recursively lists the cycles or (s, t)-paths using the classical binary partition: given an edgee in G, list all the solutions containinge, and then all the solutions not containinge, at each time modifying the graph. In order to avoid recursive calls (in the binary partition) that do not list solutions, we have used acertificate, as a data structure, whose cost for dy- namically updating is constant with respect to the number of solutions produced.
In order to prove the complexity obtained, we have exploited the properties of the binary recursion tree corresponding to the binary partition. For more details, see [2].
4 Enumerating bubbles
A DNA fragment, that is an RNA-coding sequence, is transformed in a Pre- mRNA sequence, through the transcription phase, in which sequences ofexons and sequences ofintrons alternatively occur. The removal of all the sequences of introns and of some sequences of exons leads to the mRNA sequence, that is a protein-coding sequence, that translated leads to a protein. Since not any exon is transcribed in the mRNA sequence, there can be many possible mRNA sequences. For instance, let he1, i1, e2, i2, e3, i3, e4, i4i be a fragment of DNA, where for any j, with 1 ≤ j ≤ 3, ej and ij are the j-th sequence of exons and introns respectively. The possible resulting mRNA sequences containinge1
are he1, e2, e3, e4i, he1, e2, e3i, he1, e2, e4i, he1, e3, e4i, he1, e2i, he1, e3i, he1, e4i. The underlying phenomenon is called alternative splicing and checking all the alternative events has been shown in [13] to correspond to checking recognisable patterns in a de Bruijn graph built from the reads provided by a sequencing project. The pattern corresponds to an (s, t)-bubble: an (s, t)-bubble is a pair of vertex-disjoint (s, t)-paths that only sharessandt.
Since the k-mers correspond to all words of length k present in the reads (strings) of the input dataset, and only those, in relation to the classical de Bruijn graph for all possible words of sizek, the de Bruijn graph for NGS data may then not be complete. We have ignored all the details related to the treatment of NGS data using De Bruijn graphs, and consider instead the more general case of finding all (s, t)-bubbles in an arbitrary directed graph. In particular we show the first linear delay algorithm to identify all bubbles. A previous known algorithm presented in [13] was an adaptation of Tiernan’s algorithm for cycle enumeration [15] which does not have a polynomial delay. In the worst case the time elapsed between the output of two solutions is proportional to the number of paths in the graph, i.e. exponential in the size of the graph. Our algorithm A.Marino. Algorithms for biological graphs: analysis and enumeration
is a non trivial adaptation of Johnson’s cycle enumeration algorithm [6] in a directed graph with the same theoretical complexity. Notably, the method we propose enumerates all bubbles with a given source withO(|V|+|E|) delay. The algorithm requires an initial transformation of the graph, for each sources, that takesO(|V|+|E|) time and space; this transformation reduces the enumeration of bubbles to the enumeration of constrained cycles in a special graph.
References
1. V. AcuŻna, E. Birmel´e, L. Cottret, P. Crescenzi, F. Jourdan, V. Lacroix, A. Marchetti-Spaccamela, A. Marino, P. V. Milreu, M.-F. Sagot, and L. Stougie.
Telling stories: Enumerating maximal directed acyclic graphs with a constrained set of sources and targets. Theor. Comput. Sci., 457:1–9, 2012.
2. E. Birmel´e, R. A. Ferreira, R. Grossi, A. Marino, N. Pisanti, R. Rizzi, and G. Saco- moto. Optimal listing of cycles and st-paths in undirected graphs. InSODA, pages 1884–1896, 2013.
3. P. Crescenzi, R. Grossi, M. Habib, L. Lanzi, and A. Marino. On computing the diameter of real-world undirected graphs. Theor. Comput. Sci., 514:84–95, 2013.
4. P. Crescenzi, R. Grossi, C. Imbrenda, L. Lanzi, and A. Marino. Finding the di- ameter in real-world graphs - experimentally turning a lower bound into an upper bound. InESA (1), pages 302–313, 2010.
5. P. Crescenzi, R. Grossi, L. Lanzi, and A. Marino. On computing the diameter of real-world directed (weighted) graphs. InSEA, pages 99–110, 2012.
6. D.B. Johnson. Finding all the elementary circuits of a directed graph. SIAM J.
Comput., 4(1):77–84, 1975.
7. S. Klamt and A. von Kamp. Computing paths and cycles in biological interaction graphs. BMC Bioinformatics, 10:181, 2009.
8. C. Klein, A. Marino, M.-F. Sagot, P.V. Milreu, and M. Brilli. Structural and dynamical analysis of biological networks.Briefings in functional genomics, 2012.
9. B. Lars, P. Boldi, M. Rosa, J. Ugander, and S. Vigna. Four degrees of separation.
InWebSci, pages 33–42, 2012.
10. G. Madalinski, E. Godat, S. Alves, D. Lesage, E. Genin, P. Levi, J. Labarre, J.- C. Tabet, E. Ezan, and C. Junot. Direct introduction of biological samples into a ltq-orbitrap hybrid mass spectrometer as a tool for fast metabolome analysis.
Analytical Chemistry, 80(9):3291–3303, 2008.
11. P. V. Milreu, C. Klein, L. Cottret, V. Acu˜na, E. Birmel´e, M. Borassi, C. Junot, A. Marchetti-Spaccamela, A. Marino, L. Stougie, F. Jourdan, P. Crescenzi, V. Lacroix, and M.-F. Sagot. Telling metabolic stories to explore metabolomics data: a case study on the yeast response to cadmium exposure. Bioinformatics, 30(1):61–70, 2014.
12. M. E. J. Newman. The structure and function of complex networks. SIAM RE- VIEW, 45:167–256, 2003.
13. G. Sacomoto, J. Kielbassa, R. Chikhi, R. Uricaru, P. Antoniou, M.-F. Sagot, P. Pe- terlongo, and V. Lacroix. Kissplice: de-novo calling alternative splicing events from rna-seq data. BMC Bioinformatics, 13(S-6):S5, 2012.
14. B. Schwikowski and E. Speckenmeyer. On enumerating all minimal solutions of feedback problems. Discrete Applied Mathematics, 117(1-3):253 – 265, 2002.
15. J. C. Tiernan. An efficient search algorithm to find the elementary circuits of a graph. Communonications ACM, 13:722–726, 1970.
A.Marino. Algorithms for biological graphs: analysis and enumeration
Timed process calculi:
from durationless actions to durational ones
?Marco Bernardo1 Flavio Corradini2 Luca Tesei2
1 Dipartimento di Scienze di Base e Fondamenti, Universit`a di Urbino, Italy
2 Scuola di Scienze e Tecnologie, Universit`a di Camerino, Italy
Abstract. Several timed process calculi have been proposed in the lit- erature, which mainly differ for the way in which delays are represented.
In particular, a distinction is made between integrated-time calculi, in which actions are durational, and orthogonal-time calculi, in which ac- tions are instantaneous and delays are expressed separately. To reconcile the two approaches, in a previous work an encoding from the integrated- time calculus CIPA to the orthogonal-time calculus TCCS was defined, which preserves timed bisimilarity. To complete the picture, in this pa- per we consider the reverse translation, by examining the modifications to the two calculi that are needed to make an encoding feasible, as well as the behavioral equivalence that is appropriate to preserve. We then introduce an encoding from modified TCCS to modified CIPA, and show that it can only preserve the weak variant of timed bisimilarity.
1 Introduction
Computing systems are characterized not only by their functional behavior, but also by their quantitative features. In particular, timing aspects play a funda- mental role, as they describe the temporal evolution of system activities. This is especially true forreal-time systems, which are considered correct only if the execution of their activities fulfills certaintemporal constraints.
When modeling these systems, time is represented through nonnegative num- bers. In the following, we refer toabstract time, in the sense that we use time as a parameter for expressing constraints about instants of occurrences of actions.
Unlikephysical time, abstract time permits simplifications that are convenient, on the conceptual side, to obtain tractable models.
Manytimed process calculi have appeared in the literature. Among them, we mention temporal CCS [8], timed CCS [15], timed CSP [13], real-time ACP [2], urgent LOTOS [4], CIPA [1], TPL [7], ATP [11], TIC [12], and PAFAS [6].
As observed in [10, 14, 5], these calculi differ on the basis of a number of time- related options, some of which are recalled below:
– Durationless actions versusdurational actions. In the first case, actions are instantaneous events and time passes in between them; hence, functional
?Work partially supported by the MIUR-PRIN project CINA.
behavior and time are orthogonal. In the second case, every action takes a fixed amount of time to be performed and time passes only due to action execution; hence, functional behavior and time areintegrated.
– Relative time versus absolute time. Assume that timestamps are associated with the events observed during system execution. In the first case, each timestamp refers to the time instant of the previous observation. In the second case, all timestamps refer to the starting time of the system execution.
– Global clock versuslocal clocks. In the first case, there is a single clock that governs time passing. In the second case, there are several clocks associ- ated with the various system parts, which elapse independent of each other although they define a unique notion of global time.
Moreover, for timed process calculi, there are several different interpretations of action execution, in terms of whether and when it can be delayed, such as:
– Eagerness: actions must be performed as soon as they become enabled, i.e., without any delay, thereby implying that they are urgent.
– Laziness: after getting enabled, actions can be delayed arbitrarily long before they are executed.
– Maximal progress: enabled actions can be delayed arbitrarily long unless they are involved in synchronizations, in which case they are urgent.
In this paper, we focus on two different timed process calculi obtained by suitably combining the time-related options mentioned above. More precisely, the first calculus, TCCS [8], is inspired by thetwo-phase functioning principle, according to which actions are durationless, time is relative, and there is a sin- gle global clock. In contrast, the second calculus, CIPA [1], is inspired by the one-phase functioning principle, according to which actions are durational, time is absolute, and several local clocks are present.
In [5], it was shown that some of the choices concerned with the time-related options and action execution interpretations are not irreconcilable, thus permit- ting the interchange of concepts and analysis techniques. More precisely, the different expressive power of the two considered process calculi was investigated by developing a bisimulation-semantics-preserving encoding of CIPA processes into TCCS processes for each action execution interpretation.
In this paper, we complete the previous expressiveness study by considering the reverse encoding from TCCS processes to CIPA processes, which may also be exploited for checking bisimilarity of TCCS processes more efficiently. As pointed out at the end of [5], there are several issues that need to be addressed before the reverse encoding can be established. Our first contribution is to provide a solution for each of the various problems. Our second contribution is the definition of the reverse encoding, together with a full abstraction result of this reverse encoding underweak timed bisimilarity, as opposed to the direct encoding demonstrated to be fully abstract with respect tostrong timed bisimilarity in [5].
The rest of the paper is organized as follows. In Sect. 2, we recall TCCS and CIPA. In Sect. 3, we discuss the main design decisions behind the reverse encoding. In Sect. 4, we define the reverse encoding and show that it preserves weak timed bisimilarity. Finally, in Sect. 5 we provide some concluding remarks.
M.Bernardo et al. Timed process calculi: from durationless actions to durational ones
2 Background
2.1 Preliminaries
We denote byAa nonempty set of visible actions – ranged over bya, b– and by A¯={¯a|a∈A}the set of corresponding coactions such that ¯¯a=afor alla∈A.
We useAct =A∪A¯∪ {τ}to indicate the set of all actions – ranged over byα, β – whereτ is the invisible action.
We denote by Rel a set of action relabeling functions. Each such function ϕ:Act →Act satisfiesϕ(τ) =τ andϕ(a) =ϕ(¯a) for alla∈Act\ {τ}.
We denote byT = (T,,v) a time domain such thatT∩Act =∅, which is equipped with an associative operationpossessing neutral element and a total order relationvsatisfyingt1vt2 iff there exists t0 ∈T such thatt1t0 =t2. Typical choices areT =NandT =R≥0, with the usual + and≤.
Finally, we denote byVar a nonempty set of process variables – ranged over byX, Y – whose occurrences can be free or bound by “rec”.
2.2 Durationless Actions: TCCS
We recall from [8] the syntax of TCCS. As in [5], we leave out the idling operatorδ and the weak choice operator⊕, as they have no direct counterpart in CIPA.
Definition 1. The set of process terms of the process languagePLTCCS is gen- erated by the following syntax:
P ::= 0 stopped process
| α.P action prefix
| (t).P delay prefix
| P+P alternative composition
| P|P parallel composition
| P\L restriction
| P[ϕ] relabeling
| X process variable
| recX :P recursion
whereα∈Act, t ∈N>0,L⊆A,ϕ∈Rel , and X ∈Var . We denote byPTCCS
the set of closed and guarded process terms ofPLTCCS.
Process 0 can neither proceed with any action, nor proceed through time.
Processα.P can perform instantaneous actionαand then evolves into processP;
actionαis urgent, hence time cannot progress beforeαis executed. Process (t).P evolves into processP after a delay equal tot.
Process P1+P2 represents a nondeterministic choice between processes P1
and P2, with the choice being resolved depending on whether an action of P1
orP2is executed first. Time does not resolve choices, in the sense that any initial passage of time common to P1 and P2 must be allowed without making the choice. ProcessP1|P2 describes the parallel composition of processesP1andP2, M.Bernardo et al. Timed process calculi: from durationless actions to durational ones
α.P−−→α P (t).P−t;P P1
−−→α P10 P1+P2
−−→α P10
P2
−−→α P20 P1+P2
−−→α P20 (t+t0).P−t;(t0).P
P−t;P0 (t0).Pt+t
0
−;P0 P1
−−→α P10
P1|P2
−−→α P10|P2
P2
−−→α P20
P1|P2
−−→α P1|P20
P1
−t;P10 P2
−t;P20
P1+P2
−t;P10+P20 P1
−−→a P10 P2
¯
−−→a P20
P1|P2
−−→τ P10|P20
P1
−t;P10 P2
−t;P20
P1|P2
−t;P10|P20 P−−→α P0 α /∈L∪L¯
P\L−−→α P0\L
P−t;P0 P\L−t;P0\L P−−→α P0
P[ϕ]−−→ϕ(α)P0[ϕ]
P−t;P0 P[ϕ]−t;P0[ϕ]
P{recX :P ,→X}−−→α P0 recX:P−−→α P0
P{recX:P ,→X}−t;P0 recX :P−t;P0 Table 1.Structural operational semantic rules for TCCS
where any two complementary actions may synchronize thereby resulting in aτ action; also in this case, any initial passage of time must be permitted.
Process P\L behaves as process P except for actions in L∪L, which are¯ forbidden; this operator is useful to force synchronizations between complemen- tary actions. ProcessP[ϕ] behaves as processP, with the difference that every performed action is transformed viaϕ; this operator allows processes with dif- ferent actions to communicate. Finally, recX :P represents a recursive process, which behaves as processP in which every free occurrence ofX is replaced by recX :P itself; the resulting process will be denoted byP{recX :P ,→X}.
Following [8], the intuitive meaning of process terms is formalized in Table 1.
Transition relation−−→on the left represents the functional behavior. Transition relation−; on the right represents the timing behavior according to time ad- ditivity (second and third rules) and time determinism (fourth and fifth rules);
the second rule is necessary for the applicability of the fourth and fifth ones, while the third rule is necessary for the forthcoming equivalence.
A notion of weak bisimilarity for TCCS was studied in [9]. It is an extension of Milner’s weak bisimilarity that is capable of summing up delays while abstracting fromτ actions. Weak transitions are defined as follows:
M.Bernardo et al. Timed process calculi: from durationless actions to durational ones
– ==⇒= (−−→τ )∗. – ==a⇒= ==⇒−−→a ==⇒.
– ==αˆ⇒= ==⇒ ifα=τ, ==αˆ⇒===α⇒ifα6=τ.
– ==t⇒= ==⇒−t;=1 =⇒ · · ·==⇒−tn;==⇒ wheret=P
1≤i≤nti,n∈N≥1.
Definition 2. A symmetric relationBoverPTCCSis a weak timed bisimulation iff, whenever(P1, P2)∈ B, then for all actionsα∈Act and delayst∈N>0:
– For each P1
−−→α P10 there existsP2 ˆ
==α⇒P20 such that (P10, P20)∈ B. – For each P1
−t;P10 there existsP2
==t⇒P20 such that(P10, P20)∈ B. P1≈TCCSP2 iff (P1, P2)is contained in a weak timed bisimulation.
2.3 Durational Actions: CIPA
We recall from [1] the syntax of CIPA. As in [5], we add the relabeling operator.
Definition 3. The set of process terms of the process languagePLCIPA is gen- erated by the following syntax:
Q ::= nil inactive process
| a.Q durational action prefix
| waitt.Q waiting prefix
| Q+Q alternative composition
| Q|Q parallel composition
| Q\L restriction
| Q[ϕ] relabeling
| X process variable
| recX:Q recursion
where a∈ Act\ {τ},t ∈ N>0,L ⊆A, ϕ ∈Rel , and X ∈ Var . We denote by PCIPA the set of closed and guarded process terms ofPLCIPA.
Process nil cannot proceed with any action, but can let time pass. Processa.Q can perform urgent actionaand evolves into processQafter the execution ofa has finished; all occurrences of an action are assumed to have the same duration, which is established by a function∆: (Act\{τ})→N>0such that∆(¯a) =∆(a).
Process waitt.Q waits for time t and then becomes process Q. All the other operators work as expected, with the additional constraints that each relabeling functionϕmust preserve durations, i.e.,∆(ϕ(a)) =∆(a) for all a∈Act\ {τ}, and any pair of actionsaand ¯acan synchronize only if they start at the same time, yielding aτ action with the same duration as the two original actions.
Following [1], the set KP of states correspond to process terms augmented with local clocks, so to keep track of the time elapsed in the various sequential components. The shorthandt ⇒Q means that the clock valuet ∈N≥0 is dis- tributed over all subprocesses ofQaccording to the extended syntax forKP: M.Bernardo et al. Timed process calculi: from durationless actions to durational ones
